How do I bring this face up and snap to another face?
What I use is the Bridge Edge Loops function. It's not perfect since it won't move all the faces but will connect the top edges cleanly.

Make sure you are either in Edge or Vertex Mode and select the two Edge loops you want to connect.



Press W and select Bridge Edge Loops.



In the operation settings, check Merge and set Merge Factor to 0. If 0 moves the wrong  Edge loop then set the value to 1.





Lastly you'll have to move the rest of the faces manually. It's not perfect but works well enough for my own usage. I believe this is what user277143 is referring to but as I can't add comments yet I instead created a new answer.